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Antiderivatives

- 1. Introduction Calculus is the study of change. In both of these branches (Differential and Integral), the concepts learned in algebra and geometry are extended using the idea of limits. Limits allow us to study what happens when points on a graph get closer and closer together until their distance is infinitesimally small (almost zero). Once the idea of limits is applied to our Calculus problem, the techniques used in algebra and geometry can be implemented. Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration. Leibniz’s Creation of the Calculus In the years from 1672 to 1676, spent in Paris, Leibniz’s slowly flowering mathematical genius matured. During this time, he developed the principal features and notation of his version of the Calculus. Various methods had been invented for determining the tangent lines to certain classes of curves, but as yet nobody had made known similar procedures for solving the inverse problem, that is deriving the equation of the curve itself from the properties of the tangents. Leibniz stated the inverse tangent problem thus: “To find the locus of the function, provided the locus which determines the subtangent is known.” By the middle of 1673, he had settled down to an exploration of this problem, fully recognizing that “almost the whole of the theory of the inverse method of tangents is reducible to quadratures [integrations].”
- 2. Because Leibniz was still struggling with the notation for his calculus, it is not surprising that these early calculations were clumsy. Either he expressed his results in rhetorical form or else used abbreviations, such as “omn.” For the Latin omnia (“all”) to mean “sum.” The letter l was used to symbolize what we should write as dy, the “difference” of two neighboring ordinates . Differential Calculus In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus. The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point. Integral Calculus The integral is an important concept in mathematics. Integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral is defined informally as the signed area of the region in the xy-plane that is bounded by the graph of f, the x-axis and the vertical lines x = a and x = b. The area above the x-axis adds to the total and that below the x-axis subtracts from the total. The term integral may also refer to the related notion of the antiderivative, a function F whose derivative is the given function f. In this case, it is called an indefinite integral and is written:
- 3. What is the difference between these two? Differential Calculus cuts something into small pieces to find how it changes. Integral Calculus joins (integrates) the small pieces together to find how much there is. Antidifferentiation Antidifferentiation is the process of finding the set of all the antiderivatives of a given function. The symbol ∫ denotes the operation of antidifferentiation, and we write ∫ 푓(푥)푑푥 = 퐹(푥) + 퐶 where 퐹′ (푥) = 푓(푥) and 푑(퐹(푥)) = 퐹(푥) + 퐶. The expression 퐹(푥) + 퐶 is the general antiderivative of f. Leibniz introduced the convention of writing the differential of a function after the antidifferentiation symbol. The advantage of using the differential in this manner will be apparent when we compute antiderivatives by changing the variable. From the above equations, we can write: ∫ 푑(퐹(푥)) = 퐹(푥) + 퐶 This equation states that when we antidiffentiate the differential of a function, we obtain that function plus an arbitrary constant. So we can think of the ∫ symbol for antidifferentiation as meaning that operation which is the inverse of the operation denoted by d for computing differential. If { 퐹(푥) + 퐶} is the set of all functions whose differentials are푓(푥)푑푥, it is also the set of all functions whose derivatives are f(x). Antidifferentiation, therefore, is considered as the operation of finding the set of all functions having a given derivative. Notation for Antiderivatives The antidifferentiation process is also called integration and is denoted by the symbol ∫ called an integral sign. The symbol ∫ 푓(푥)푑푥 is called the indefinite integral of f(x), and it denotes the family of antiderivatives of f(x). That is, if 퐹′ (푥) = 푓(푥) for all x, then ∫ 푓(푥)푑푥 = 퐹(푥) + 퐶 where f(x) is called the integrand and C the constant of integration. The differential dx in the indefinite integral identifies the variable of integration. That is, the symbol ∫ 푓(푥)푑푥 denotes the “antiderivative of f with respect to x” just as the symbol dy/dx denotes the “derivative of y with respect to x.”
- 4. Integration Tables Basic Integration Formulas Rules Antiderivatives 1. Constant Rule ∫ 푑푥 = 푥 + 퐶 2. Simple Power Rule ∫ 푥 푛 푑푥 = 푥 푛+1 푛 + 1 + 퐶, 푛 ≠ −1 3. General Power Rule ∫ 푢푛 푑푢 푑푥 푑푥 = ∫ 푢푛 푑푢 = 푢푛+1 푛 + 1 + 퐶, 푛 ≠ −1 4. Simple Log Rule ∫ 1 푥 푑푥 = ln|푥| + 퐶 5. General Log Rule ∫ 1 푢 푑푢 푑푥 푑푥 = ∫ 1 푢 푑푢 = ln|푢| + 퐶 6. Simple Exponential Rule ∫ 푒푥 푑푥 = 푒푥 + 퐶 7. General Exponential Rule ∫ 푒푢 푑푢 푑푥 푑푥 = ∫ 푒푢 푑푢 = 푒푢 + 퐶 Trigonometric Functions Integration Formulas Functions Antiderivatives 1. ∫ 푐표푠푥 푑푥 푠푖푛푥 + 퐶 2. ∫ 푠푖푛푥 푑푥 −푐표푠푥 + 퐶 3. ∫ 푠푒푐2 푥 푑푥 푡푎푛푥 + 퐶 4. ∫ 푐푠푐2 푥 푑푥 −푐표푡푥 + 퐶 5. ∫ 푠푒푐푥푡푎푛푥 푑푥 푠푒푐푥 + 퐶 6. ∫ 푐푠푐푥푐표푡푥 푑푥 −푐푠푐푥 + 퐶
- 5. Sample Problems Solving the Antiderivatives of Basic Equation
- 6. Sample Problems Solving the Antiderivatives of Trigonometric Functions
- 7. Differential Equations A differential equation is an equation involving a function and one or more of its derivatives. For instance, 3 푑푦 푑푥 − 2푥푦 = 0 is a differential equation in which y = f(x) is a differentiable function of x. A function y = f(x) is called a solution of a given differential equation if the equation is satisfied when y and its derivatives are replaced by f(x) and its derivatives. For example, 푦 = 푒−2푥 is a solution to the differential equation 푦′ + 2푦 = 0 because 푦′ = −2푒−2푥 , and by substitution we have −2푒−2푥 + 2푒−2푥 = 0 Furthermore, we can readily see that 푦 = 2푒−2푥 , 푦 = 3푒−2푥 , 푦 = 1 2 푒−2푥 are also solutions to the same differential equation. In fact, the functions 푦 = 퐶푒−2푥 where C is any real number, are all solutions. We call this family of solutions the general solution of the differential equation 푦′ + 2푦 = 0. A particular solution of a differential equation is any solution that is obtained by assigning specific values to the constants in the general solution. Geometrically, the general solution of a given differential equation represents a family of curves, called solution curves one for each value assigned to the arbitrary constants.
- 8. Example of Differential Equations (First and Second Order) 1. Show that 푑2 푦 푑푥2 = 2 푑푦 푑푥 a solution of y = c1 + c2e2x Solution: Since y = c1 + c2e2x, then: 푑푦 푑푥 = 2c2e2x and 2 푑푦 푑푥 = 4c2e2x Therefore, 푑2 푦 푑푥2 = 2 푑푦 푑푥 . References Books Burton, David M. The History of Mathematics Larson and Hostetler. Brief Calculus with Applications. “Integration.” Fourth Printing. P 316-518 Leithold, Louis. TC 7 (The Calculus 7). P 315-32 Websites http://en.wikipedia.org/wiki/Calculus http://en.wikipedia.org/wiki/Integral http://en.wikipedia.org/wiki/Differential_calculus http://www.mathsisfun.com/calculus/integration-introduction.html http://tutorial.math.lamar.edu/Classes/CalcII/IntegralsWithTrig.aspx